There are different kinds of disability. For SSDI, the fee is fixed by the SSA and is something like $6000 max (may have gone up this year). If you're at the hearing stage for SSDI, a lawyer or advocate firm (my ins. had me use Allsup for example) would probably be a good idea. They are only paid if you win. Maybe you shouldn't try it without representation of some sort unless you've got totally solid documentation from your Drs. - including Residual Functional Capacity forms and all of that.
There are other kinds of disability ins. For private policies, the ins. carrier could be sued for damages. I assume most lawyers like these cases.
For ERISA cases (what most people get thru their work), the laws are very complicated and slanted towards the carrier.
This is a specialty area and a lawyer should prob. be involved in the appeal. Except for special circumstances, no new info can be introduced after the appeal is submitted. It's my understanding that these cases are usually done on contingency - the lawyer takes the risk but a fee like 30% is typical. It's also my understanding that it's for the duration of the payments from the carrier.